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Stirling’s approximation
Stirling’s formula gives an approximation for , the factorial function. It is
We can derive this from the gamma function. Note that for large ,
| (1) |
where
with . Taking and multiplying by , we have
| (2) |
Taking the approximation for large gives us Stirling’s formula.
There is also a big-O notation version of Stirling’s approximation:
| (3) |
We can prove this equality starting from (2). It is clear that the big-O portion of (3) must come from , so we must consider the asymptotic behavior of .
First we observe that the Taylor series for is
But in our case we have to a vanishing exponent. Note that if we vary as , we have as
Related:
MinkowskisConstant, AsymptoticBoundsForFactorial
Synonym:
Stirling's formula, Stirling's approximation formula
Type of Math Object:
Theorem
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
68Q25 Analysis of algorithms and problem complexity30E15 Asymptotic representations in the complex domain
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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